Grade 6 Mathematics · Unit 7: Integers, Rational Numbers, and the Coordinate PlaneStandard 6.NOS.9

Lesson 7.6Determine Distance on the Coordinate Plane

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can find the distance between two points on the coordinate plane using absolute value.

Why This Matters Captain Vega has two treasure locations on her map: a buried chest at (-3, 2) and a hidden cave at (4, 2). Both are at the same height (y = 2) on the map. She needs to figure out how far apart they are so she knows how much rope to bring. Another pair of treasures is at (1, -3) and (1, 5) — same column, different rows. How far apart is each pair?

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
DistanceSpanish: Distancia How far apart two points are. It is never negative. always a positive length
Absolute valueSpanish: Valor absoluto How far a number is from zero. It is never negative. |−7| = 7
Horizontal distanceSpanish: Distancia horizontal How far apart two points are going left or right. uses the x-coordinates
Vertical distanceSpanish: Distancia vertical How far apart two points are going up or down. uses the y-coordinates
Coordinate planeSpanish: Plano cartesiano A grid with a line going across and a line going up to plot points. x-axis across, y-axis up
IntegerSpanish: Número entero Whole numbers and their opposites, like -2, -1, 0, 1, 2. ..., -3, -2, -1, 0, 1, 2, 3, ...

How it worksWorked example

These numbers are not on your problems. The steps are. Follow them with your own numbers.

Each unit is 1 meter. Find the distance from P(−9, 4) to Q(3, 4), and from P to R(−9, 11).

  1. P and Q share y = 4, so use x. −9 and 3 have opposite signs: 9 + 3 = 12 m.
  2. P and R share x = −9, so use y. 4 and 11 are both positive: 11 − 4 = 7 m.

Answer: PQ = 12 m and PR = 7 m

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

Now let's find the distance from (2, 3) to (2, 8). What do these points share? The same x = 2, so it is a vertical line.

Answer:

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

  • The two points share the -coordinate, so the segment runs .
  • I used absolute value because .
  • The points sit on sides of the axis, so I the two distances.
  • The distance is units, which means on this map.

Word bank distanceabsolute valuehorizontalverticalcoordinate planeaxisunitsaddsubtractx-coordinatey-coordinatesegment

Watch out!A common mistake

A common mistake in Distance on the Coordinate Plane is subtracting the coordinates as if both points were on the same side of zero, instead of adding their absolute values when the points are on opposite sides. For example, to find the distance between (-6, 3) and (2, 3), a student might subtract 6 − 2 = 4, but since -6 and 2 are on opposite sides of zero, the absolute values must be added instead: |-6| + |2| = 6 + 2 = 8. The correct distance is 8 units, not 4. Before you submit, ask: 'Are these two points on opposite sides of zero

Grade 6 Mathematics · Unit 7: Integers, Rational Numbers, and the Coordinate PlaneStandard 6.NOS.9

Lesson 7.6Determine Distance on the Coordinate Plane

Level 0Most support · every hint printed

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can find the distance between two points on the coordinate plane using absolute value.

Why This Matters Captain Vega has two treasure locations on her map: a buried chest at (-3, 2) and a hidden cave at (4, 2). Both are at the same height (y = 2) on the map. She needs to figure out how far apart they are so she knows how much rope to bring. Another pair of treasures is at (1, -3) and (1, 5) — same column, different rows. How far apart is each pair?

How it worksWorked example

These numbers are not on your problems. The steps are. Follow them with your own numbers.

Each unit is 1 meter. Find the distance from P(−9, 4) to Q(3, 4), and from P to R(−9, 11).

  1. P and Q share y = 4, so use x. −9 and 3 have opposite signs: 9 + 3 = 12 m.
  2. P and R share x = −9, so use y. 4 and 11 are both positive: 11 − 4 = 7 m.

Answer: PQ = 12 m and PR = 7 m

Part 1Understand the ideamodels and meaning
  1. 1Complete the table. Show how you found each value.

    Both points in each row share a coordinate. Find the distance.

    Point 1Point 2Shared CoordinateDistance
    (-5, 1)(3, 1)y = 1
    (2, -4)(2, 6)x = 2
    (-3, -2)(-3, 4)x = -3
    (-1, 5)(4, 5)y = 5
    Steps to try
    1. Each row shares a coordinate, so every distance is a straight horizontal or vertical count.
    2. Subtract the coordinates that DIFFER and take the absolute value. If the signs differ, add the two absolute values.
    3. Row 1: the y's match, so count from x = -5 to x = 3. Go -5 to 0, then 0 to 3, and add the two counts.
    Show your work
Part 2Apply itreal-world problems
  1. 2Circle the letter of the best answer. Show how you know.

    What is the distance between (-4, 3) and (2, 3)?

    1. A2 units
    2. B4 units
    3. C6 units
    4. D8 units
    Steps to try
    1. Both points have y = 3, so they sit on the same horizontal line — only the x-values differ.
    2. Find the gap between x = -4 and x = 2. The path crosses zero, so add the two distances |-4| and |2|.
    3. Count from -4 to 0 (that's 4 units), then from 0 to 2. Add the two parts together.
    Show your work
  2. 3Circle the letter of the best answer. Show how you know.

    What is the distance between (5, -2) and (5, 4)?

    1. A2 units
    2. B4 units
    3. C6 units
    4. D10 units
    Steps to try
    1. Both points share x = 5 — the segment is VERTICAL, so compare the y-values -2 and 4.
    2. The path crosses zero: split it into a below-zero part and an above-zero part using absolute values.
    3. From y = -2 up to 0 is 2 units; from 0 up to 4 is 4 more. Add the parts.
    Show your work
  3. 4Circle the letter of the best answer. Show how you know.

    A rectangle has vertices at (1, 1), (5, 1), (5, 4), (1, 4). What is its perimeter?

    1. A8 units
    2. B12 units
    3. C14 units
    4. D20 units
    Steps to try
    1. Perimeter is the distance all the way around — you need the width AND the height first.
    2. Width: subtract the x-values 5 and 1. Height: subtract the y-values 4 and 1. Then add all four sides.
    3. Find 5 − 1 for the width and 4 − 1 for the height. Now double each and add — that covers all four sides.
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
    1. 1Name itWhich measure is asked?
    2. 2Write the formulaBefore any numbers.
    3. 3SubstituteMatch each letter.
    4. 4Unitunits, sq units, or cubic?

Explain your thinkingHow did you use coordinates and absolute value to find the distance between two points in the same row or column?

Sentence starter The points ___ and ___ share the same ___ -coordinate. To find the distance, I found |___| + |___| = ___, which equals ___ units.

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help